In which article/book chapter did Cantor, Hilbert, and Poincaré formally define or directly discusse the term “potential infinity”?

Some media sources say that "Cantor claimed that there would only be potential infinity, not actual infinity"

In addition, the following link claims that Hilbert, Poincaré, and Cantor were fully aware of the distinction between "potential infinity" and the actual one.

Is there a formal distinction between potential and actual infinities?

Of course all those mathematicians talked a lot about infinity in their publications; however I am having trouble finding an exact quote on "potential infinity" from them. In which chapters did they mention the exact term "potential infinity"?

I did notice that Fraenkel mentioned "potential infinity" a lot, but not for Hilbert or Poincaré.

• Welcome to HSM. The article you have linked in your first sentence appears to be full of inaccuracies and falsehoods. For example, it states "For instance, Cantor proved that the set of all functions from $(0,1)$ →$\mathbb N$ is countable. Thus he defined a one-to-one and onto function from the interval $(0,1)$ to natural numbers." This is plainly false, and not entirely coherent. Also, the statement you attribute to Cantor in the first sentence of your post is, according to the article, the view of Aristotle rather than Cantor.
– nwr
Jun 26, 2020 at 21:38
• @Nick I agree, that part is definitely incorrect. Jun 26, 2020 at 22:40

Hilbert devoted a whole paper to infinity, On the infinity, published in 1925 (1).

Hilbert was one of the most tenacious defenders of the "paradise of the infinity" revealed by Cantor's work. His paper was aimed at 'saving' the Cantorian set theory approach to infinity from the paradoxes and antinomies generated by actual infinity during the so-called foundation crisis of mathematics.

In On the infinity he makes a short account of the concept of infinity in analysis, set theory, physics and logic, and he speaks explicitly of potential and actual infinity:

If we wanted to characterize briefly the new conceptions of the infinity that Cantor introduced, we could no doubt say: in analysis we deal with the infinitely small or the infinitely large only as a limit notion - as something that is becoming, coming to be, being produced - that is, as we say, with the potential infinity. But this is not the real infinity itself. That we have when, for example, we consider the totality of the numbers $$1, 2, 3, 4,...$$ itself as a completed entity, or when we regard the points of a line segment as a totality of objects that is actually given and complete. This kind of infinity is called actual infinity. (2)

On the contrary, Poincaré doesn’t accept Cantor’s actual infinity. He addressed the problem of Cantor’s infinity in his speech L’avenir des mathématiques at the International Conference of Mathematicians, held in Rome in 1908, in which he states that

“Cantorism” was “a good pathological case”, which could make happy the physician called to treat it.(3)

Poincaré devotes several papers to infinity, his last article about the subject is La logique de l’infinie, published in 1909. He in particular speaks of actual infinity in La mathématique e la logique (1905):

It has been the belief in the existence of actual infinity which has generated these impredicative definitions […] There insn’t such a thing as actual infinity: Cantorians forgot it and fell into contradictions. (4)

(1) Hilbert, On the infinity (1925), in J. van Heijenoort ed., From Frege to Gödel, 1967, Cambridge University Press.

(2) Ibid., p.373.

(3) U. Bottazzini, Infinito, Il Mulino, 2018, p. 254, my translation.

(4) Poincaré, La mathématique e la logique (1905), p. 55 http://tomlr.free.fr/Math%E9matiques/Fichiers%20Claude/Les%20Mathematiques%20Et%20La%20Logique.pdf. (my translation).

You have misinterpreted the article you refer to; nowhere does it say that "Cantor claimed that there would only be potential infinity, not actual infinity". In fact, it says the opposite:

Furthermore, Cantor claimed that we could add and multiply infinity sets. Until that time, humans had followed Aristotle’s ideas about infinity. According to Aristotle, if we multiply the number 3 by infinity, it would be infinite again. Infinity would swallow everything. Based on this, he claimed that there would only be potential infinity, not actual infinity.

It's Aristotle claiming that there is only potential infinity, and we have since learned that in many ways, Aristotle was just a big windbag.

A potential infinity is the kind we work with all the time in calculus: the limit of something as something else controlling it it is allowed to grow with out bound. There is no "largest" natural number, you can always add one and get a bigger one. In a set-theoretical framework with basic axioms, any proper class is a potential infinity.

An absolute infinity is a "completed" one. As Cantor later showed, you need a whole new axiom to make it work formally. The usual way to do this is to assert the existence of a complete set of natural numbers ($$\omega$$) as an object you can manipulate. It's this Axiom of Infinity that formally defines an absolute infinity.

A number of years ago I was led to the infinite real whole numbers [die unendlichen realen ganzen Zahlen] without having realized that they were concrete numbers of real significance.$$^1$$

Cantor would never have claimed there was only potential infinity. His transfinite numbers are all completed ones.

Joseph Warren Dauben's biography of Cantor discusses this concept in Chapter 5, which is all about Cantor's 1883 Grundlagen einer allgemeinen Mannigfaltigskeitlehre.This title may cause you to do a double-take, as it transtates as "Foundations of a Theory of Manifolds". Here is the quote you're looking for:

As far as the mathematical infinite is concerned: to the extent that it has found justifiable use in science and made a useful contribution, the mathematical infinite has principally occurred in the meaning of a variable magnitude, either growing beyond all limits or diminishing to an arbitrary smallness, always, however remaining finite. I call this infinite the non-genuine-infinite [das Uneigentlich-unendliches].$$^2$$

There was great opposition to Cantor's ideas by people who couldn't wrap their heads around them. There's a remarkable quote later on in the book, criticizing "proofs" that completed infinities couldn't exist, because they assumed there were only potential infinities to begin with:

All so-called proofs against the possibility of actually infinite numbers are faulty, as can be demonstrated in every particular case, and as can be concluded on General grounds as well. It is their πρώτον ψεύδος [first lie] that from the outset they expect or even impose all the properties of finite numbers upon the numbers in question, while on the other hand the infinite numbers, if they're to be considered in any form at all, must (in their contrast of the finite numbers) constitute an entirely new kind of number, whose nature is entirely dependent upon the nature of things and is an object of research but not of our arbitrariness or prejudices.

-Cantor to Eneström$$^3$$

$$^1$$:Georg Cantor: His Mathematics and Philosophy of the Infinite, by Joseph Warren Dauben, Princeton University Press, 1990, quotation, p. 57

$$^2$$Cantor, G. Grundlagen einer allgemeinen Mannigfaltigskeitlehre. Ein mathematisch-philosophischer Versuch in der Lehre des Unendlichen. Leipzig, B.G. Teubner, 1883. Later in Mathematische Annalen. Translated by Uwe Parpart, first appearing in The Campaigner, January-February 1976, Vol 9 Nos. 1-2, P. 70.

$$^3$$ Dauben, quotation, p. 125